The article that I spoke of relates to more than a single study. I think the study you a referencing is one that received a great deal of attention last week.

Another straw-man that is often set up is the one that places all non-project-based math practice into the category of mindless, manipulation of symbols. Other types of practice involve meta-cognition, categorization, and synthesis. This strengthens students' ability to "figure things out."

As time allows, I will review that research you mentioned.

John AndersonLowell, IN

________________________________From: John Clement <clement@hal-pc.org>To: math-learn@yahoogroups.comSent: Thursday, May 5, 2011 11:04 AMSubject: Re: [math-learn] Other Comments On High School Algebra 2

As I recall the article on "minimal guidance instruction" set up a straw manand proceeded to burn it. What they called minimal guided instruction doesnot in my experience exist. Perhaps a few misguided individuals do it, butnone of the PER techniques do this. They also did not consoder any of thePER research. This article has been, in my opinion, debunked. Look at thework of Shayer & Adey or the Hellers at U. Minn. Then there is the Benezitexperiment.

While complex open ended activities may inhibit some transfer to long-termmemory, short practice tends to make students think that it is just a matterof memorizing without understanding. The subject of long-term memory ismuch more complex than just transfer of information. It is a dynamic systemwhere recall changes existing ideas through the process or reconsolidation.When ideas are merely practiced, without cognition, the pre-existingparadigms are still there and come out to bite you.

For years people in physics have gotten students to memorize Newton's lawsand gave them simple practice problems. But when asked about a real lifesituation which involved applying them, their "natural" paradigmsoverwhelmed their memorize info, and they got them wrong. I know thatstudents have practiced invert and multiply, but still in college they willtell me that 1/s divided by s is 1. Then I ask what is 1/2 divided by 2 andthey still say 1. So then I ask them to visualize 1/2 of something anddivide it by 2 and they get the correct 1/4.

As to building rigor incrementally, that is a "false" idea. Rigor is oftenequated with the ability to do a variety of specific tasks, but in realitywhat the student needs to gain is enough ability to use cognition to figureout things, or be able to go to a source to find what they need to solve aproblem. Then if they are to become mathemeticians they will gain highproficiency by practicing their trade.

John M. ClementHouston, TX

> > Complex problems certainly play an important role in > mathematical understanding. When those problems develop > rigor incrementally, much is to be gained. However, I > believe a balance should be struck. Educational > psychologists point to the limited ability of complex, > open-ended activities to move understanding into the > long-term memory of students. One such article, Why Minimal > Guidance During Instruction Does Not Work: An Analysis of the > Failure of Constructivist, Discovery, Problem-Based, > Experiential, and Inquiry-Based Teachingby Kirschner, > Sweller, and Clark does a good job of summarizing and > clarifying this understanding.>